Finite group actions on surfaces: A Riemann surface approach to classification by genus

Abstract

Actions of finite groups on compact surfaces can be studied by fixing the surface and asking which groups are able to act on it. The present work addresses this question by classifying the effective actions of finite groups on compact surfaces according to their genus. The central idea is to pass from the differentiable to the holomorphic setting: averaging an arbitrary Riemannian metric yields a G-invariant one, which determines an almost-complex structure on the surface. A central result establishes that every such structure arises from a unique Riemann surface structure so then the elements of the group act as holomorphic or antiholomorphic automorphisms. The required theory of Riemann surfaces is developed, with special attention to uniformization and the Riemann-Hurwitz formula. In this way, the problem is reduced to understanding automorphisms. This is carried out in each of the three cases determined by the genus, where the geometry of the surface dictates how much symmetry is possible. In genus zero the surface admits a large amount of symmetry, whereas from genus one onwards the admissible groups become increasingly rigid, culminating in the bound that governs the high genus case.